Cross-layer scheduling has been proposed to effectively boost the spectral efficiency of multi-user orthogonal frequency division multiple access (OFDMA) systems through multi-user selection diversity under two important assumptions users are delay insensitive and channel state information at the transmitter (CSIT) is perfect. However, in practice, users may be delay sensitive with heterogeneous delay requirements, and CSIT usually becomes outdated in time varying channel. When CSIT is outdated, there will always be systematic packet errors even if powerful channel coding is applied, resulting in significant degradation on the delay performance of heterogeneous users. In this paper, we propose a novel delay-sensitive cross-layer design framework for multi-user OFDMA systems with heterogeneous delay requirements in slow fading channels, by utilizing both queueing theory and information theory to model the system dynamics as well as accounting impacts upon outdated CSIT. Based on the framework, the optimal power and subcarrier allocation solution is shown to maintain heterogeneous users' delay constraints and guarantee a fixed target outage probability, and its asymptotic multi-user diversity gain over fixed allocation scheme is also obtained analytically. Simulation results further show that our proposed delay-sensitive CSIT error considerate schemes provide robust system performance enhancement over naive scheduler while satisfying heterogeneous delay requirements even at moderate to high CSIT errors.